Question

Find a parametric representation for the surface. (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of u and/or v.)
The plane through the origin that contains the vectors i − j and j − k

Answer 1

I got
\[r=<u,v,-u-v>\]

the computer didn't accept it though

Answer 2

did you for for \(\vec r=<\vec u,\vec v,−\vec u−\vec v>\) just because this is \(\pi: \ x+y+z = 0\)??

Answer 3

help pwease

Answer 4

yea thats what I did

Answer 5

Sorry I did the cross product to get r=i+j+k then subed in the values for x y z

Answer 6

i see where you are coming from and perhaps the problem is with data entry as often happens on websites but maybe you could also approach it differently, more simply even, using the basic info they have given you.

Info given = "The plane through the origin that contains the vectors i − j and j − k"

so you know that any point (x,y,z) can be found by starting at the origin (0,0,0) and using the given vectors, ie
\(<x,y,z> = <0,0,0> + u <1,-1,0> + v<0,1,-1> \)
\( = u <1,-1,0> + v<0,1,-1>\)

for example, we know (2,3,-5) is on plane, thus testing

\[\left(\begin{matrix}2 \\ 3 \\-5 \end{matrix}\right) = u \left(\begin{matrix}1 \\ -1 \\0 \end{matrix}\right) + v \left(\begin{matrix} 0 \\ 1 \\-1 \end{matrix}\right) \]

\(\implies u = 2, v = 5\)

so for "(Enter your answer as a comma-separated list of equations." you might try:

x = u, y = - u + v, z = -v

i think you'd be fine with a human marking system :p